It is a well known fact (see here for instance), that under reasonable conditions any curve on a smooth manifold can be realized as a geodesic for some given connection.

The natural construction when $\gamma$ is closed, for example, consists in to realize a Riemannian metric in the form: $$g = fg_1 + hg_2,$$ where $g_2$ is a flat Riemannian metric and $\{f,h\}$ is a unit partition. In this construction, $\gamma$ is a geodesic for the metric $g_2$.

I was wondering: is there a manner to make such analogous construction preserving positive curvature? Or at least, for some planes?

Imagine you have a Riemannian manifold $(M,\tilde g)$ such that $\tilde g$ has positive sectional curvature and a smooth curve $c$ on $M$. Is it possible to obtain a Riemannian metric $g$ (possible related to $\tilde g$) on $M$ in which $c$ is a geodesic and the curvature of $g$ is also positive?

If we try obtain a Riemannian metric with positive sectional curvature following the construction on the second paragraph, then I imagine a natural way to proceed is using the arbitrarily of $g_1$ to impose further conditions on the unity partition. I elaborate:

$K_{hg_2}$ by the formulae of conformal change only depends on the Hessian of the function $h$. Since $h$ has compact support it has directions where the Hessian can be negative and this can make the curvature of $g$ be negative. But we also have the curvature of $fg_1$. It seems naîve, but couldn't we just search for: we ask that $g_1$ be such that when the Hessian of $h$ is negative (for example when $h$ is decreasing) then since $\{f,h\}$ is a unity partition, $f$ is increasing, then it has positive Hessian, we ask that the curvature of $g_1$ is positive on such directions, and where $f$ has negative Hessian, then we ask $K_{g_1}$ ti be negative. I have many suspicious that this doesn't work although I cannot justify.

I appreciate any hep


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